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Derivation of Black-Scholes formulaTseng, Cho-Ming 07 December 2009 (has links)
The Black-Scholes European option pricing formula can be derived in several ways. In this dissertation we present several methods that can be used to derive
this formula, including partial differential equation method, the risk-neutral pricing method, the martingale measure method, and the change of numeraire technique
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Analytic Approaches to the Pricing Black-Scholes Equations of Asian OptionsYu, Wei-Hau 05 July 2012 (has links)
Asian option is an option which payoff depends on the average underlying price over some some specific time period. Although there is no closed form solution of asian option, appropriate change of variable and Num¡¦eraire would reduce some terms of equation satisfies the Asian call price function. This thesis presents asian option¡¦s properties and process of reduction terms.
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The technique of measure and numeraire changes in optionShi, Chung-Ru 10 July 2012 (has links)
A num¡¦eraire is the unit of account in which other assets are denominated.
One usually takes the num¡¦eraire to be the currency of a country.
In some applications one must change the num¡¦eraire due to the finance considerations.
And sometimes it is convenient to change the num¡¦eraire because
of modeling considerations. A model can be complicated or simple, depending
on the choice of thenum¡¦eraire for the method.
When change the num¡¦eraire, denominating the asset in some other unit of account,
it is no longer a martingale under ˜P . When we change the num¡¦eraire,
we need to also change the risk-neutral measure in order to maintain risk
neutrality.
The details and some applications of this idea developed in this thesis.
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Growth optimal portfolios and real world pricingRamarimbahoaka, Dimbinirina 12 1900 (has links)
Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / In the Benchmark Approach to Finance, it has been shown that by taking the
Growth Optimal Portfolio as numéraire, a candidate for a pricing derivatives
formula under the real world probability can be given. This result allows
us to price in an incomplete financial market model. The result comes from
two different approaches. In the first approach we use the supermartingale
property of portfolios in units of the benchmark portfolio which leads to the
fact that an equivalent measure is not needed. In the second approach the
numéraire property of the Growth Optimal Portfolio is used. The numéraire
portfolio defines an equivalent martingale measure and by change of measure
using the Radon-Nikodým derivative, a real world pricing formula is derived
which is the same as the one given by the first approach stated above.
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Management portfolia s několika referenčními aktivy / Portfolio Management with Multiple BenchmarksNavrátil, Robert January 2017 (has links)
Portfolio Management with Multiple Benchmarks Bc. Robert Navrátil Abstract: In this thesis, we study a maximal volatility portfolio that treats all assets in a symmetric way and related option contract. To preserve symmetry we need numeraire that treats all assets symmetrically. We choose market index with equal weights. In case of two assets we focus on a variation of a passport option on the portfolio. The optimal strategy for the investor is the mentioned maximal volatility portfolio. We extend the known optimal strategy for the option to a richer class of convex payoff functions. We also show a modification of the optimal strategy for maximizing the probability of ending above or at a desired level. We later extend the symmetric market model to case of three assets, which can be even further extended to an arbitrary number of assets. The three asset model requires more parameters than are observable from the data, however we show indistinguishably of the model on the choice of parameters under very natural conditions. Both numerical simulations and an application on real data is provided. 1
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Three essays on valuation and investment in incomplete marketsRinger, Nathanael David 01 June 2011 (has links)
Incomplete markets provide many challenges for both investment decisions and valuation
problems. While both problems have received extensive attention in complete markets,
there remain many open areas in the theory of incomplete markets. We present the results
in three parts. In the first essay we consider the Merton investment problem of optimal portfolio
choice when the traded instruments are the set of zero-coupon bonds. Working within
a Markovian Heath-Jarrow-Morton framework of the interest rate term structure driven by
an infinite dimensional Wiener process, we give sufficient conditions for the existence and
uniqueness of an optimal investment strategy. When there is uniqueness, we provide a characterization
of the optimal portfolio. Furthermore, we show that a specific Gauss-Markov random field model can be treated within this framework, and explicitly calculate the optimal
portfolio. We show that the optimal portfolio in this case can be identified with the discontinuities
of a certain function of the market parameters. In the second essay we price a claim,
using the indifference valuation methodology, in the model presented in the first section. We
appeal to the indifference pricing framework instead of the classic Black-Scholes method due
to the natural incompleteness in such a market model. Because we price time-sensitive interest
rate claims, the units in which we price are very important. This will require us to take
care in formulating the investor’s utility function in terms of the units in which we express the
wealth function. This leads to new results, namely a general change-of-numeraire theorem in
incomplete markets via indifference pricing. Lastly, in the third essay, we propose a method
to price credit derivatives, namely collateralized debt obligations (CDOs) using indifference.
We develop a numerical algorithm for pricing such CDOs. The high illiquidity of the CDO
market coupled with the allowance of default in the underlying traded assets creates a very
incomplete market. We explain the market-observed prices of such credit derivatives via the
risk aversion of investors. In addition to a general algorithm, several approximation schemes
are proposed. / text
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